Restaurants create menus to control costs and ensure efficiency in the kitchen. But they're very aware that service and satisfaction are what keep customers coming back. Many chain restaurants offer an extensive, standard menu that's loaded with flexibility, so if your off-menu request is reasonable and the kitchen has the ingredients to make it, the chef will most likely honor it.
With independent restaurants, it's hit-and-miss. Some offer a fluid menu that draws on seasonally available foods. Even if the restaurant wants to honor your request, the chef is limited by what's in the kitchen. But if you're there on a slow night, you could ask the chef to get creative. It gives you a chance to try something new -- and the chef gets an opportunity to test new inventions.
Customer requests help shape the menu. If your favorite dish dropped off the new menu, that doesn't mean it's not available. It could just mean that it wasn't a high selling item, and the manager put a more popular dish in its place. Chances are the chef still remembers how to cook your special dish -- and will, if you ask for it.